I've answered all kinds of algebra problems in various ways, depending on the learning style of my student. For example, 2x(3x-6) = 30 can be easily solved...
Unless we need to back up and go over using negative numbers, fractions, or distribution in more detail, this is how to solve: 2x(3x-6) = 30 6x^2 - 12x = 30 6x^2 - 12x - 30 = 0 Now, one needs to decide whether this particular problem can be factored for a solution or whether the quadratic formula should be used to solve instead. This particular problem cannot be factored and must be solved with the quadratic formula: x = (12± √ ( (-12)^2 - 4(6)(-30) ) ) / (2(6)) x = (12± √ ( 144 - 4(-180) ) / 12 x = (12± √864) / 12 x ≈ -1.45, 3.45
Find the derivative of -2x^2 + 4x + 6
This is a very basic derivative problem that can identify issues with the basic understanding of finding a derivative or the basic algebra one must know in order to proceed with calculus. The answer is -4x + 4 so if my student has a different answer, I see how he/she arrived at that answer in order to figure out what to explain and in how much detail. Then, my student can understand how to find the correct answer on this, then a similar problem.
Find the inverse of the composition of functions f(x) = 2x and g(x) = x + 4
First, one must find the composition g ∘ f: (g ∘ f )(x) = 2x + 4 (this step can be explained in more detail as needed) Then reverse this process to find the inverse, so subtract 4, and then divide by 2: (g ∘ f)^(−1)(x) = (x − 4)/2 Working through this with a student reveals what exactly needs to be explained in more detail, including any fundamental algebra concepts required.